Assigning quantum labels to variationally computed rotational-vibrationaleigenstates of polyatomic molecules
Edit Mátyus,1 Csaba Fábri,1 Tamás Szidarovszky,1 Gábor Czakó,1,2 Wesley D. Allen,3
and Attila G. Császár1,a�
1Laboratory of Molecular Spectroscopy, Institute of Chemistry, Eötvös University,P.O. Box 32, H-1518 Budapest 112, Hungary2Department of Chemistry and Cherry L. Emerson Center for Scientific Computation,Emory University, Atlanta, Georgia 30322, USA3Department of Chemistry and Center for Computational Chemistry, University of Georgia,Athens, Georgia 30602, USA
�Received 12 January 2010; accepted 20 May 2010; published online 20 July 2010�
A procedure is investigated for assigning physically transparent, approximate vibrational androtational quantum labels to variationally computed eigenstates. Pure vibrational wave functions areanalyzed by means of normal-mode decomposition �NMD� tables constructed from overlapintegrals with respect to separable harmonic oscillator basis functions. Complementary rotationallabels JKaKc
During the past decade, remarkable progress has beenachieved in the development of “numerically exact” varia-tional methods for computing rotational-vibrational eigen-states of polyatomic molecules.1–9 Notwithstanding the im-proved capabilities for converging on energy levels, theassignment and interpretation of the multitudinous resultingwave functions, especially at higher energies, remains achallenge.10–13 The problem is often exacerbated by the useof sophisticated basis sets and coordinate representations.While an unambiguous labeling of molecular rovibrationalstates is helpful in the physical interpretation of measuredspectra, it is required for the construction of spectroscopicdatabases.14–16 Different investigations often employ differ-ent labels for the same quantum states or spectroscopic tran-sitions, confounding efforts to compile self-consistent data-bases.
An ideal labeling scheme would be physically incisiveand independent of the coordinates and basis functions usedto represent the Hamiltonian and the wave function. How-ever, assignment schemes can be very useful even if theserequirements are not fully met. Among the techniques thathave been employed in the analyses of variationally com-puted nuclear-motion wave functions are “node counting”along specified cuts of coordinate space,17,18 the determina-
tion of “optimally separable” coordinates,10,19–28 the use ofnatural modal representations,18,29 and the evaluation of co-ordinate expectation values.17 An alternative approach to as-signing molecular eigenstates is provided by effectiveHamiltonian methods, particularly in relatively low-energyregions.
The canonical models of the vibrations and rotations of amolecule are the quantum mechanical harmonic oscillator�HO� �Ref. 30� and rigid-rotor �RR� �Ref. 31� approxima-tions, respectively. The low-lying states of semirigid mol-ecules have traditionally been described by labels based onmultidimensional normal-mode vibrational wave functionsconjoined with RR rotational wave functions represented in asymmetric-top basis. A widespread preference for RRHO la-bels persists both for the appealing simplicity of the under-lying models and for historical reasons. Of course, theRRHO labeling scheme is inherently model dependent, un-like methods based on natural modals, for example. Varia-tional vibrational computations have often3,32–45 employedthe Eckart–Watson Hamiltonian expressed in normalcoordinates,46–48 which leads straightforwardly to a HO la-beling of the lower-lying eigenstates. During the more than30-year development of variational nuclear motion computa-tions with exact kinetic energy operators, the emphasis hasgradually shifted away from the Eckart–Watson Hamiltonianto Hamiltonians expressed in internal coordinates.2,4–9,18,49–53
Nevertheless, for molecular systems of medium size �morethan four atoms but less than eight�, a special role will bea�Electronic mail: [email protected].
THE JOURNAL OF CHEMICAL PHYSICS 133, 034113 �2010�
maintained for the Eckart–Watson Hamiltonian, perhaps ex-pressed in a discrete variable representation �DVR�,54–57 us-ing potential energy functions given in internal coordinatesand a multidimensional HO basis.
This paper presents an easily automated protocol for la-beling variational rovibrational wave functions by construct-ing certain standardized types of normal-mode decomposi-tion �NMD� and rigid-rotor decomposition �RRD� tables.The NMD labeling scheme formalizes normal-mode repre-sentations that have long been in use for variational vibra-tional wave functions but have been underutilized for quan-titative assignments. Our RRD approach is more novel, andit is amenable to rovibrational computations using any set ofvibrational coordinates. The NMD and RRD procedures aredemonstrated here by variational rovibrational computationsfor the H2O, HNCO, trans-HCOD, NCCO, and H2CCO mol-ecules.
II. ROTATIONAL-VIBRATIONAL LABELINGPROTOCOL
The rotational-vibrational labeling protocol leading toNMD and RRD tables was implemented in our own nuclearmotion program system called DEWE.3 DEWE employs aDVR54,56 of the complete Eckart–Watson Hamiltonian,46–48 abasis set composed of Hermite-DVR functions,55,58 and a fullpotential energy surface �PES� expressed in arbitrary coordi-nates. DEWE computes the required eigenvalues and eigen-functions iteratively.59,60 The vibrational part of the DEWE
program is described in detail in Ref. 3. The treatment hasbeen extended during the present work to include rotations aswell.
Let us consider the nJth rovibrational wave function�nJ
J �Q ,� ,� ,�� as a linear combination of rotational-vibrational basis functions,
�nJ
J �Q,�,�,�� = �i=1
N
�L=1
2J+1
cnJ,iLJ �i�Q�RL
J��,�,�� , �1�
where �� ,� ,�� is the usual set of Euler angles, Q= �Q1 ,Q2 , . . . ,Q3M−6� denotes the normal coordinates of anM-atomic molecule, J is the rotational quantum number, andRL
J�� ,� ,�� denotes the Wang-transformed symmetric-top ro-tational basis functions31 indexed by L. The vibrational basisfunctions �i�Q� are assumed to be products of one-dimensional functions in each vibrational degree of freedom,and N=N1N2¯N3M−6 is the total size of the multidimen-sional vibrational basis.
A. Normal-mode decomposition of vibrations
A pure vibrational state �m�Q� �J=0� can be describedas a linear combination of product functions of HOs,
�m�Q� = �i=1
N
Cm,i�iHO�Q� . �2�
Due to the normalization of the wave function and the ortho-normality of the basis functions, �i=1
N �Cm,i�2=1 and one canwrite
Cm,i = ��iHO��mQ. �3�
The �Cm,i�2 coefficients are, from now on, referred to as theelements of the NMD table.
The labeling of “exact” vibrational wave functions�m�Q� with HO quantum numbers can be accomplished bypicking out the dominant contributors in Eq. �2�, which canbe read directly from a NMD table. This simplification issimilar in spirit to that employed during potential energydistribution, kinetic energy distribution, or total energy dis-tribution analyses61–66 of harmonic vibrations executedwithin the GF formalism30 to describe normal modes viainternal coordinates.
The quantum analog67 of the Kolmogorov–Arnold–Moser68 theorem provides the basis for assigning quantumnumbers via separability approximations, like the normal-mode model. A NMD coefficient larger than 0.5 means aclose similarity of the exact, nonseparable wave function tothat provided by the separable HO Hamiltonian. A smallercoefficient does not mean that no good approximate quantumnumbers can be found—it simply means that the HO ap-proximation may not provide the best separation. This studyis not concerned with searching for better separations thanprovided by the HO approximation ubiquitous in molecularspectroscopy.
Obviously, it would be advantageous to be able to pro-duce NMDs from arbitrary wave functions represented witharbitrary basis functions and coordinates. In general, the in-tegral given in Eq. �3� might be computed by numericalquadratures as
Cm,i = �j
wj�iHO�� j��m�� j� , �4�
where wj and � j are appropriately chosen quadrature weightsand points, respectively, in the multidimensional space, andreal-valued functions are assumed. However, if the varia-tional wave functions are computed by programs built uponthe use of internal coordinates, the computation of NMDs ishindered considerably as the internal coordinate and the HOwave functions whose overlap must be computed are basedon different ranges and volume elements. Computation ofNMDs is not at all simple in this case, and singularitieswhich might arise in the Jacobi determinant could providefurther difficulties.
B. Assignment and rigid-rotor decomposition ofrotations
For the eigenstates of the field-free rovibrational Hamil-tonian, the J rotational quantum number is exact, while thewidely used Ka and Kc labels are approximate and corre-spond to �K� for the prolate and oblate symmetric-top limitsof the RR,31 respectively. In the present subsection a two-step algorithm based on certain nonstandard overlap integralsis proposed to match the computed rovibrational states withpure vibrational states and then generate the Ka and Kc la-bels.
By rearranging Eq. �1�, one obtains
034113-2 Mátyus et al. J. Chem. Phys. 133, 034113 �2010�
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�nJ
J �Q,�,�,�� = �L=1
2J+1
RLJ��,�,���
i=1
N
cnJ,iLJ �i�Q��
= �L=1
2J+1
RLJ��,�,���nJL
J �Q� . �5�
From now on, �nJLJ �Q� will be referred to as the Lth vibra-
tional part of �nJ
J �Q ,� ,� ,��. Because the eigenfunctions ofthe rotational-vibrational Hamiltonian are orthonormal, theoverlap of a vibration-only wave function �m�Q� and a rovi-brational wave function �nJ
J �Q ,� ,� ,�� �J�0� is alwayszero, and thus not useful for making assignments. A way tocircumvent this problem is to introduce the overlap of theLth vibrational part of �nJ
J �Q ,� ,� ,�� and the vibration-only�m�Q� as
SnJL,mJ = ��nJL
J �Q���m�Q�Q
= �i=1
N
�j=1
N
cnJ,iLJ Cm,j��i�Q��� j�Q�Q
= �i=1
N
cnJ,iLJ Cm,i, �6�
where the integration is carried out over the 3M−6 vibra-tional coordinates, and an orthonormal vibrational basis andreal linear combination coefficients are assumed. SnJL,m
J pro-vides a measure of the similarity of �nJL
J �Q� and �m�Q�: thelarger the magnitude of SnJL,m
J , the more similar the vibra-tional parts of the two functions are. The next step is to sumthe absolute squares of the SnJL,m
J quantities with respect to L,
PnJ,mJ = �
L=1
2J+1
�SnJL,mJ �2 = �
L=1
2J+1 ��i=1
N
cnJ,iLJ Cm,i�2
. �7�
After converging M J=0 and NJ J�0 eigenstates byvariational procedures, NJM square-overlap sums are com-puted over all of the J=0 and J�0 pairs. The quantitiesPnJ,m
J �nJ=1,2 , . . . ,NJ and m=1,2 , . . . ,M� can be regardedas elements of a rectangular matrix with NJ rows and Mcolumns. For a given J, those �2J+1� �nJ
J �Q ,� ,� ,�� rovi-brational states belong to a selected �m�Q� pure vibrationalstate which give the 2J+1 largest PnJ,m
J values. This means ofidentification is valuable because the rovibrational levels be-longing to a given vibrational state appear neither consecu-tively nor in a predictable manner in the overall eigenspec-trum.
It is important to emphasize the pronounced dependenceof the quantities PnJ,m
J on the embedding of the body-fixedframe, as exhibited in Eqs. �5� and �7�. The DEWE code em-ploys the Eckart frame,46 which is expected to be a trenchantchoice for the overlap calculations due to a minimalizedrovibrational coupling. Of course, this rotational labelingscheme can be extended to other variational rovibrationalapproaches employing arbitrary internal coordinates and em-beddings.
After assigning 2J+1 rovibrational levels to a pure vi-brational state, the next step is to generate the Ka and Kc or
=Ka−Kc labels. Such assignments could be naively basedon the canonical energy stacking of asymmetric-top JKaKcstates, derived from the symmetric-top limits, the symmetrylabels of the states, and the noncrossing rule.31 A rigorousapproach is to set up what we call RRD tables. The twoapproaches do not necessarily give the same labels, althoughthis problem occurred in only one case during the presentstudy investigating low-J states. In order to compute theRRD coefficients it is necessary to evaluate the overlap inte-gral
SnJ,m,mJ
J = ��nJ
J �Q,�,�,����m�Q� · mJ
J ��,�,��Q,�,�,�
= �L=1
2J+1
�i=1
N
cnJ,iLJ �
M=1
2J+1
�k=1
N
Cm,k · CmJ,MJ · ��i�Q���k�Q�Q
· �RLJ��,�,���RM
J ��,�,���,�,�
= �L=1
2J+1
�i=1
N
cnJ,iLJ · Cm,i · CmJ,L
J �8�
between the nJth rovibrational state and the product of themth vibrational state and mJth RR eigenfunction. The RRcomponent of the product is given by a linear combination ofthe Wang functions RL
J with expansion coefficients CmJ,LJ ,
mJ
J ��,�,�� = �L=1
2J+1
CmJ,LJ RL
J��,�,�� . �9�
Note that the notation employed does not restrict the sum-mation by symmetry; thus, certain blocks of the CmJ,L
J coef-ficients will necessarily be zero. Recognizing that these co-efficients are elements of a unitary matrix, the quantities inEqs. �7� and �8� are connected by the condition
PnJ,mJ = �
mJ=1
2J+1
�SnJ,m,mJ
J �2. �10�
Because the �m�Q� ·mJ
J �� ,� ,�� functions form an orthonor-mal basis of dimension N�2J+1�, it is also obvious that
�m=1
N
�PnJ,mJ �2 = 1. �11�
In light of these relationships, we define the RRD coeffi-cients as the absolute square of the overlaps �SnJ,m,mJ
J �2, andarrange them in a rectangular table whose rows are the exactstates under consideration, �nJ
J �Q ,� ,� ,��, and whosecolumns are the above-defined “basis” states,�m�Q�mJ
J �� ,� ,��.
III. NUMERICAL EXAMPLES
After developing eigenstate labeling capabilities into ourcode DEWE, applicable to semirigid molecules of arbitrarysize, the utility of the proposed NMD and RRD protocolswas investigated for examples of three-, four-, and five-atomic molecules—H2O, HNCO, trans-HCOD, NCCO, andH2CCO. Important details of the calculations performed anddefinition of the normal coordinates for all the species inves-tigated, including the appropriate transformation matrices
034113-3 Assignment of rovibrational states J. Chem. Phys. 133, 034113 �2010�
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and reference structures, are given in the supplementarymaterial.69
A. Normal-mode decomposition tables
1. NMD of water
Assigning the large number of computed �and measured�rovibrational states of water up to its first dissociation limit isan extremely demanding task.17 Without valid assignments,however, there is no hope of extending the information sys-tems characterizing water spectroscopy beyond what is avail-able at present.14
Our NMD analysis of H216O �Table I� is based on the
PES of Refs. 70 and 71. All of the vibrational eigenstates upto 7000 cm−1, including the fundamentals,72 are describedaccurately by the normal-mode model. The concept ofpolyads12 and the polyad number P, defined here as P=2�v1+v3�+v2, have often been used to analyze the vibra-tional states of water �see, e.g., Ref. 73�. Large NMD coef-ficients ��97%� are found for the ground state �P=0� andthe bending fundamental �P=1�, both one-dimensionalblocks. For the three-dimensional P=2 manifold ��2−�4�,the strongest mixing occurs outside the polyad block; n.b.
2�1 and �1+�3 contribute 10% and 9% to �3 and �4, re-spectively. This mixing would likely be diminished in a natu-ral modal representation. Unlike the P=2 case, the largestmixings for the P=3 states ��5−�7� are found within thepolyad block, in accord with the usual polyad arguments.Among the P=4 states ��8−�13�, �11 is the most stronglymixed, the largest components therein being 2�1�48%� and�1�11%�. In fact, none of the diagonal NMD values for �11,�12, and �13 exceeds 70%, indicating that the normal-modepicture has already started to break down for the purelystretching part of the P=4 polyad. Thus, the NMD analysisnicely documents the anticipated transformation fromnormal-mode to local-mode behavior. Finally, we note thatthe NMD values in Table I are in full agreement with thecoefficients in Eq. �33� of Whitehead and Handy,33 despitethe use of completely different PESs in the two studies. SuchNMD transferability across PESs and computational method-ologies is a merit for interpreting vibrational spectra.
Overall, as compared to later examples, for the low-energy states considered H2
16O provides a well-behaved ex-ample for the NMD analysis, supported also by the fact thatthe frequency order of the exact states corresponds to that ofthe harmonic basis states. For higher energies, above about
TABLE I. The lowest-energy part of the NMD table of H216O.
aRows of variational vibrational wave functions ��i� with energy levels are decomposed in terms of columns of HO basis states with reference energy levels�. NMD coefficients in percent; energies in cm−1 relative to the corresponding variational or harmonic zero-point vibrational �ZPV� level appearing in row1 or column 1, respectively.bThe decomposition was extended to 80 states in each row and column; � values denote the corresponding sums of the NMD coefficients over these states.Computed from the CVRQD PES of Refs. 70 and 71. Twenty basis functions were used for each vibrational degree of freedom. The nuclear masses mH
=1.007 276 5 u and m16O=15.990 526 u were adopted.
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12 000 cm−1,17 mixing of the basis states becomes so pro-nounced that the normal-mode labels lose any simple physi-cal meaning. Nevertheless, the quantitative characterizationprovided by the NMD array remains useful in uniquely iden-tifying vibrational eigenstates derived from diverse sources.
2. NMD of tetra-atomic molecules
Application of the NMD procedure to three tetra-atomictest cases, HNCO, trans-HCOD, and NCCO, provides thearrays in Tables II–IV. Technical details related to the chosenPESs are given in supplementary material.69 Isocyanic acid�HNCO� is a classic quasilinear molecule whose spectros-copy has been extensively studied and whose anharmonicforce field was first computed by one of us in Ref. 74. TheNCCO, trans-HCOH, and trans-HCOD molecules have re-cently been isolated and characterized for the first time, aidedby selected NMD data we have previously reported.75,76 Theexamples collected in this section show that strong mixing ofnormal-mode wave functions at low vibrational energies isnot a rare exception, even for fundamentals. Interestingly, themixing can become so strong that the very notion of a fun-damental vibrational state becomes ill defined. Besides itstheoretical delicacy, this behavior has practical conse-quences. For instance, the strong mixing is manifested in thedistorted intensity pattern in the case of the 14N13C12C16Oisotopologue �Table IV, vide infra�.76
The NMD array for HNCO �Table II� includes the 12wave functions lying below 1750 cm−1 in relative energy.Seven of these wave functions have leading NMD values of�80%. In particular, the � 5 , 6 , 4� bending fundamentals��1−�3� at �577,659,777� cm−1 have diagonal NMD coef-ficients of �94, 99, 89�%, respectively, making these assign-ments very clear. Likewise, the �2 5 , 5+ 6 , 4+ 6� bendingovertone and combination levels ��4 ,�6 ,�9� at�1143,1271,1473� cm−1 have diagonal NMD elements of�80, 90, 88�%, in order. In stark contrast, the states��5 ,�7 ,�8� lying at �1263,1325,1354� cm−1 involve astrong Fermi resonance triad of the 2�6, �3, and �4+�5
basis states. It is striking how ambiguous the identification ofthe 3 symmetric N–C–O stretching fundamental is, as the�3 basis state is the largest contributor to both �5 and �7.The best assignments for ��7 ,�8� would appear to be� 3 ,2 6�, with contributions from the ��3 ,2�6� basis func-tions of �46, 59�%, respectively. However, the only remain-ing possibility for 4+ 5 would then become �5, and the�4+�5 NMD coefficient for this wave function is only 22%,which is third largest in the list. In brief, an intricate structureis revealed for the vibrational eigenstates of HNCO in themid-IR region that would be poorly understood withoutNMD as a quantitative tool.
For deuterated trans-hydroxymethylene (trans-HCOD�,NMD data are reported in Table III for a total of 21 vibra-tional wave functions lying below 2900 cm−1 in relative en-
TABLE II. The lowest-energy part of the NMD table of HNCO.
aSee footnote a to Table I.bObtained with an all-electron CCSD�T�/cc-pCV5Z quartic internal coordinate force field taken from Ref. 78. Seven basis functions were used for eachvibrational degree of freedom. The decomposition was extended to 100 states in each row and column; � values denote the corresponding sums of the NMDcoefficients over these states. Atomic masses, in u, mH=1.007 825, m14N=14.003 074, m12C=12, and m16O=15.994 915 were adopted.
034113-5 Assignment of rovibrational states J. Chem. Phys. 133, 034113 �2010�
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ergy. The first 12 wave functions, including those for thefundamental levels 3, 4, 5, and 6, have dominant diago-nal NMD values ��90%�. Thus, all vibrational states lyingbelow 2400 cm−1 are remarkably well described by thenormal-mode picture. In contrast, the higher vibrationalstates appearing in Table III show substantial mixing in theNMD array. For �13 at 2627 cm−1, �2 contributes 83%,which is still sufficient to clearly identify this state as the 2
�OuD stretch� fundamental. However, in attempting to as-sign the remaining fundamental � 1�, we find that �15 at2683 cm−1 is 45% �1+37% ��3 + �4�, whereas �17 at2730 cm−1 is 33% �1+51% ��3 + �4�. These NMD datareveal that the CuH stretching fundamental is in strongFermi resonance with a combination level involving theHuCuO bending and CuO stretching vibrations. Whilethe best assignment for 1 �CuH stretch� is 2683 cm−1, onemust accept that the corresponding wave function �15 con-tains less than 50% of this vibrational character.
The NMD results for the 14N13C12C16O isotopologue ofthe carbonyl cyanide radical are given in Table IV for thelowest 16 vibrational wave functions, all lying below950 cm−1 in relative energy. The CwN and CvO stretch-ing fundamentals, 1�a��=2170 cm−1 and 2�a��=1853 cm−1, respectively, that were computed in our earlierstudy,76 lie outside the energy region considered in Table IV.Large diagonal NMD coefficients ��95%� allow the 5�a��, 6�a��, and 4�a�� bending fundamentals to be readily as-signed to wave functions �1, �2, and �6 at 219, 262, and567 cm−1, respectively. Nonetheless, identification of the re-maining CuC stretching fundamental � 3�a��� suffers fromthe same type of ambiguity seen above for HNCO and trans-HCOD. In particular, �10 at 777 cm−1 and �12 at 795 cm−1
exhibit a strong Fermi resonance between �3 and the com-bination level �4+�5. The apparent CuC stretching funda-mental is �12, if assigned on the basis of the 55% �3 contri-bution. However, this choice would mean that the CuC
TABLE III. The lowest-energy part of the NMD table of trans-HCOD.
aSee footnote a to Table I.bObtained with the all-electron CCSD�T�/cc-pCVQZ quartic internal coordinate force field taken from Ref. 75. Nine basis functions were used for eachvibrational degree of freedom. The decomposition was extended to 40 states in each row and column; � values denote the corresponding sums of the NMDcoefficients over these states. Atomic masses, in u, mH=1.007 825, mD=2.014 102, m12C=12, and m16O=15.994 915 were adopted.
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stretch in 14N13C12C16O is shifted +9.1 cm−1 relative to thecorresponding wavenumber in the parent isotopologue. Inother words, a counterintuitive blueshift occurs upon substi-tution of a heavier carbon isotope, as discussed in Ref. 76,illustrating the intricacies that Fermi resonances can engen-der. The strong mixing also manifests itself in the computed“intensity stealing” between �10 and �12, as documented inTable IV.
3. NMD of ketene
In previous years the five-atomic ketene molecule�H2CCO� was too large for adequate variational nuclear mo-tion treatments. This proved to be quite unfortunate becauseketene exhibits several peculiar spectroscopic features, assummarized in Refs. 79–81. Some of the complexities in thelower end of the high-resolution rovibrational spectrum ofketene arise because the three lowest fundamentals cluster inthe 430–610 cm−1 region and the next two fundamentalsoccur in the 960–1120 cm−1 window. Understanding the en-suing resonances, assigning their spectral signatures, andtreating them with theoretical techniques encounter severe
difficulties. Thus, it is no surprise that “the rich history ofinfrared and microwave studies of the ketene molecule is amicrocosm of the development of modern spectroscopy”�Ref. 79�. The NMD and RRD tables generated in this studyserve well the purpose of unraveling the complex spectros-copy of this simple molecule.
Table V presents the NMD table of ketene for vibrationalstates up to 1520 cm−1 in relative energy. The underlyingvariational computations are based on the local PES of Ref.79. The vibrational states up to 1050 cm−1 exhibit little mix-ing and have dominant NMD coefficients of �91%. How-ever, most of the states in the 1050–1550 cm−1 windowhave much smaller leading NMD coefficients due to anhar-monic resonances. The wave functions ��8 ,�9 ,�10,�11� ly-ing at �1071,1113,1169,1211� cm−1 involve a complicated�2�6 ,�5+�6 ,�4 ,2�5� Fermi resonance tetrad that cloudsthe assignment of the CvC stretching fundamental � 4�. Astriking manifestation is that the �5+�6 basis state contrib-utes between 12% and 45% to all variational wave functionsin the set ��8−�11�. Our current NMD results differ substan-tially from the more approximate coefficients extracted in
TABLE IV. The lowest-energy part of the NMD table of 14N13C12C16O.
aSee footnote a to Table I.bObtained with the all-electron ROCCSD�T�/cc-pCVQZ quartic internal coordinate force field taken from Refs. 76 and 77. Nine basis functions were used foreach vibrational degree of freedom. The decomposition was extended to 160 states in each row and column; � values denote the corresponding sums of theNMD coefficients over these states. Atomic masses, in u, m14N=14.003 074, m13C=13.003 355, m12C=12, and m16O=15.994 915 were adopted.cVibrational intensities corresponding to excitations from the ZPV level, I /km mol−1, were obtained with the DEWE program and an AE-ROCCSD�T�/cc-pCVTZ third-order dipole field �Ref. 76�.
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Ref. 79, attesting to the intricacies of the vibrational mixingin this region. Nevertheless, both studies concur in the as-signment of the experimental band82 at 1116.0 cm−1 to the 4 fundamental. The 3 �CH2 scissoring� fundamental is alsostrongly mixed, in this case due to a resonance between��4 ,�8+�9� basis states, which contribute �43%, 50%� and�50%, 45%� to ��13,�15�, respectively. Therefore, the ketenemolecule provides multiple examples in which the assign-ment of vibrational fundamentals is blurred. A much moredetailed discussion of our variational vibrational computa-tions on ketene will be presented in a forthcoming paper.
B. Rigid-rotor decomposition tables
According to the protocol of Sec. II B, the vibrationalpart of a rovibrational wave function �nJ
J can be character-ized by computing the quantities PnJ,m
J of Eq. �7� derivedfrom overlap integrals with pure vibrational wave functions�m. In addition, a RRD of �nJ
J is provided by the coefficients�SnJ,m,mJ
J �2, where SnJ,m,mJ
J is the overlap defined in Eq. �8�.
Tables of RRD coefficients lead directly to KaKc labels forasymmetric tops. Overall, our protocol for variational com-putations assigns 2J+1 clearly labeled rovibrational levels toeach of the pure �J=0� vibrational states.
In our scheme the complete rovibrational label includesthe irreducible representation �irrep� � of the molecular sym-metry �MS� group, the total rotational angular momentumquantum number �J�, Ka and Kc values corresponding to theasymmetric RR, and the normal-mode vibrational quantumnumbers �v1 ,v2 , . . . ,v3M−6�. It is worth emphasizing that thefirst two labels, � and J, are exact, as they are valid for theexact nonintegrable Hamiltonian, while the last labels, Ka,Kc, and �v1 ,v2 , . . . ,v3M−6�, are inexact designations arisingfrom the approximate rovibrational Hamiltonian.
Once a corresponding pure vibrational wave function �m
is identified for a rovibrational wave function �nJ
J , one couldattempt to make the �Ka ,Kc� assignment by assuming ca-nonical energy ordering of asymmetric-top rotational states.While this approach seems to be valid most of the time, itbreaks down occasionally due to resonances. Mislabeling is
TABLE V. The lowest-energy part of the NMD table of ketene �H2CCO�.
aSee footnote a to Table I.bObtained with a quartic internal coordinate force field taken from Ref. 79. Seven and six basis functions were used for the bending- and stretching-typevibrational degrees of freedom, respectively. The decomposition was extended to 35 states in each row and column; � values denote the corresponding sumsof the NMD coefficients over these states. Atomic masses, in u, mH=1.007 825, m12C=12, and m16O=15.994 91 were adopted.
034113-8 Mátyus et al. J. Chem. Phys. 133, 034113 �2010�
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TABLE VI. Overlap quantities PnJ,mJ �Eqs. �7� and �10�� for making assignments of the first 56 J=3 rovibrational states of H2
16O by correspondence with thefirst eight pure vibrational �J=0� states.
a�: Labels of the irreducible representations corresponding to the MS group C2v�M�. J: rotational quantum number. Ka ,Kc: approximate quantum numbers ofthe asymmetric rigid rotor. Vib.: vibrational assignment based on the NMD table. ZPV=zero-point vibrational level.bNuclear masses mH=1.007 276 5 u and m16O=15.990 526 u were adopted as well as the Eckart frame specified in the supplementary material. v and rv:variational vibrational and rovibrational energy levels in cm−1 obtained with DEWE using 15 basis functions in each vibrational degree of freedom and theCVRQD PES of Refs. 70 and 71.
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often averted by first determining the proper irrep of the MSgroup for a rovibrational state and applying symmetry rulesfor �Ka ,Kc� before invoking energy ordering. For example,within the 1 vibrational state of water, the �Ka ,Kc� valuesmust be ��even, even�, �odd, odd�, �even, odd�, �odd, even��when �= �A1 ,A2 ,B1 ,B2�, in order. In contrast, for the 3 vi-brational state of water, the �Ka ,Kc� values must be ��odd,even�, �even, odd�, �odd, odd�, �even, even�� when �= �A1 ,A2 ,B1 ,B2�. These and similar rules can be built intothe automatic labeling protocol. Nonetheless, RRD coeffi-cients must be employed to establish �Ka ,Kc� assignmentsmore rigorously. The entries in an RRD table not only reflectthe proper symmetries but also quantify the mixing of theRR functions in the exact rovibrational wave functions.
Our protocol was tested by obtaining complete rovibra-tional labels and constructing RRD tables for the low-lyingrovibrational states of the parent isotopologues of water andketene over several J values. Assignments based on the PnJ,m
J
quantities are shown in Tables VI and VII for the first 56 and28 J=3 states of H2
16O and ketene, respectively. Visual rep-resentations of our results up to J=6 are provided in Fig. 1for H2
16O, and analogous depictions for ketene are given up
to J=3 in Fig. 2 �the numerical values used to generate Figs.1 and 2 are given in the supplementary material69�. Finally,example RRD tables for manifolds of J=6 states of H2
16Oand J=3 states of ketene are presented in Tables VIII and IX.
The need for an effective labeling protocol for rovibra-tional states can be readily appreciated even by a quick lookat the panels of Figs. 1 and 2. The intermingling of rotationallevels corresponding to different vibrational states alwaysoccurs whenever J�3. For example, the seven J=3 rota-tional levels of H2
16O corresponding to the 1 fundamentalspan the 8429–8574 cm−1 interval, which also containsthree rotational levels belonging to 3 that lie between 8528and 8565 cm−1. The scrambling of rovibrational states ofH2
16O is especially evident in panels �e� and �f� of Fig. 1.The PnJ,m
J diagnostics �Eqs. �7� and �10�� in Tables VIand VII are frequently very close to their ideal, unmixedvalues of 1.00 and are almost always greater than 0.90, thusproviding unambiguous quantum labels. Nevertheless,prominent exceptions sometimes occur due to resonances,which our protocol identifies successfully. For example,Table VI reveals strong mixing between JKaKc
states belong-ing to different combination levels of H2
16O: the
TABLE VII. Overlap quantities PnJ,mJ �Eqs. �7� and �10�� for making assignments of the first 28 J=3 rovibrational states of ketene �H2CCO� by correspon-
dence with the first four pure vibrational �J=0� states.
v 6831.98 7269.05 7365.97 7435.49 Rovibrational labela
a�: Labels of the irreducible representations corresponding to the MS group C2v�M�. J: rotational quantum number. Ka, Kc: approximate quantum numbers ofthe asymmetric rigid rotor. Vib.: vibrational assignment based on the NMD table. ZPV=zero-point vibrational level.bAtomic masses mH=1.007 825 u, m12C=12 u, and m16O=15.994 915 u were adopted as well as the Eckart frame specified in the supplementary material. v
and rv: vibrational and rovibrational energy levels in cm−1 obtained with DEWE using seven and six basis functions in the bending-type and stretching-typevibrational degrees of freedom, respectively, and the quartic force field of Ref. 79.
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rovibrational eigenstate at 10 177.6 cm−1 is 79%�330� 1
+ 2��+20%�322� 2+ 3��, while that for 10 182.8 cm−1 is20%�330� 1+ 2��+80%�322� 2+ 3��. This pronounced reso-nance causes a switching in relative energy of the 330 and 331
levels of 1+ 2 relative to the expected RR energy ordering�E�331��E�330��, although the difference is less than0.5 cm−1. This case is the only one encountered for H2
16O inthe current study for which the canonical sequence of RRlevels is not obeyed.
Expected near degeneracies are manifested in the assign-ments given in Tables VI and VII. Within the bending vibra-tional states of water �0 v2 0�, the JJ1−JJ0 rotational split-ting decreases uniformly from 0.20 cm−1 for v2=0 to
0.10 cm−1 for v2=3, evidencing the formation of incipientlocal-mode pairs. The ketene molecule is very nearly asymmetric top, with �A0 ,B0 ,C0� close to�9.410,0.343,0.331� cm−1, in order.79 Accordingly, a neardouble degeneracy for all values of Ka�1 is seen in therovibrational levels in Table VII.
The RRD results in Tables VIII and IX demonstrate thatthe mixing among the RR wave functions is comfortablysmall. In all cases but one in these tables there is a dominantsquare-overlap coefficient of �92%, the exception occurringfor the B2 state of H2
16O at 7821.3 cm−1 that is83%�634�2 1��+16%�624� 1+ 3��. The computed RRD mix-ing does depend slightly on the choice of rotational constantsfor the RR basis functions. The smallest mixing occurs if oneemploys the rotational constants of the corresponding purevibrational state. Nevertheless, even if alternate choices aremade, such as the equilibrium rotational constants, the in-crease in mixing is not sufficient to create ambiguity in theassignment protocol.
IV. SUMMARY
Powerful variational methods are increasingly used forcomputing accurate rovibrational states of polyatomic mol-ecules. Standard, automated procedures are needed for as-signing and interpreting the large collections of eigenstates
(a) J = 1
(b) J = 2
(c) J = 3
(d) J = 4
(e) J = 5
(f) J = 6
FIG. 1. Visualization of the correspondence between rovibrational states�J�0� and pure vibrational states �J=0� of H2
16O, ordered in columns androws, respectively, in terms of increasing energy. Dark squares match theJ�0 states with their J=0 counterparts, according to the PnJ,m
J quantities inEq. �7�. The more similar the vibrational parts, the darker the square is in thefigure. The variational computations utilized the DEWE code �Ref. 3�,adopted the Eckart frame specified in the supplementary material �Ref. 69�,and are based on the CVRQD PES �Refs. 70 and 71�.
(a) J = 1
(b) J = 2
(c) J = 3
FIG. 2. Visualization of the correspondence between rovibrational states�J�0� and pure vibrational states �J=0� of ketene, ordered in columns androws, respectively, in terms of increasing energy. Dark squares match theJ�0 states with their J=0 counterparts, according to the PnJ,m
J quantities inEq. �7�. The variational computations utilized the DEWE code �Ref. 3�,adopted the Eckart frame specified in the supplementary material �Ref. 69�,and are based on the quartic force field of Ref. 79.
034113-11 Assignment of rovibrational states J. Chem. Phys. 133, 034113 �2010�
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resulting from state-of-the-art variational computations in or-der to solve chemical problems and to compile self-consistent spectroscopic databases, inter alia.
In this paper we demonstrate the use of NMD tables as astandard protocol for making quantitative assignments of ex-act, anharmonic vibrational states. Such tables follow natu-rally in variational approaches based on the Eckart–WatsonHamiltonian, and they are applicable, in principle, to generalcoordinate representations of the vibrational problem. In-deed, NMD tables can be most beneficial when coordinatesystems are employed that are not chemically motivated.
A formalism for assigning rotational quantum labels tovariational eigenstates with J�0 is needed to complementthe NMD approach for J=0 states. We have proposed amethod that uses a diagnostic evaluated from sums ofsquares of overlap integrals to match each rovibrational levelwith a pure vibrational state. After these identifications aremade, RRD tables are utilized to find correspondences withasymmetric-top rotational eigenfunctions, and hence to as-cribe labels. Our RRD protocol is applicable regardless ofwhether the vibrational part of the wave function is describedwell by normal-mode wave functions, and it is also able toclearly identify resonances between different rotational lev-els of different vibrational states.
As tests of our procedures for labeling and analyzingrovibrational eigenstates, variational computations were per-formed on several carefully selected molecules, namely,H2
16O, HNCO, trans-HCOD, 14N13C12C16O, and ketene�H2CCO�. The NMD and RRD protocols were implementedin the DEWE program package3 employing the Eckart frameand normal coordinates corresponding to the actual PESused.
An interesting conclusion of the numerical results ob-tained is that the normal-mode picture of the vibrationalbands commonly breaks down even for some of the funda-mentals of molecules. Considerable mixing among some ofthe low-energy states seems to be the rule rather than theexception for the cases studied here. This finding may alsobe important for Eckart–Watson Hamiltonian-based vibra-tional self-consistent-field �VSCF� spectroscopy �and treat-ments based on such VSCF ansätze�. Along with symmetryclassification, the approximate vibrational and rotational la-bels attached to variationally computed rotational-vibrationaleigenstates provide information much needed by experimen-tal spectroscopists. NMD and RRD tables appear to be usefulalso for spectroscopic perturbation theory as they give a clearindication of extensive mixings among states which could beimportant when setting up the effective Hamiltonians used to
TABLE VIII. RRD tables, based on Eq. �8�, for the J=6 rovibrational states of H216O corresponding to the 12th pure vibrational state, showing all
J �Q ,� ,� ,�� are decomposed in terms of columns of RR rotational eigenfunctions �JKaKc,mJ
J �� ,� ,��� attachedto the pure vibrational states, �m�Q�, lying at v. RRD coefficients in percent, rounded to the nearest integer; energies in cm−1 relative to the ZPV energy. ARRD array is given for each of the C2v�M� symmetry blocks �A1 ,A2 ,B1 ,B2�.bThe variational rovibrational computations were performed as specified in footnote a to Table VI. The J=6 RR eigenfunctions were constructed from thevibrationally averaged rotational constants of the v=7201.2 cm−1 state: A=26.3466 cm−1, B=14.1044 cm−1, and C=8.9108 cm−1 �Ref. 83�.
034113-12 Mátyus et al. J. Chem. Phys. 133, 034113 �2010�
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interpret high-resolution spectroscopic experiments. Thus,NMD and RRD tables, resulting in complete rovibrationallabels, should be routinely computed at least for semirigidmolecules.
ACKNOWLEDGMENTS
The research in Hungary was carried out with the finan-cial support of the Scientific Research Fund of Hungary�Grant No. OTKA K72885�. The work at the University ofGeorgia was supported by the U.S. Department of Energy,Office of Basic Energy Sciences, Combustion Program�Grant No. DE-FG02-97ER14748�. This work was per-formed as part of the Task Group of the International Unionof Pure and Applied Chemistry �IUPAC� �Project No. 2004-035-1-100� on “A database of water transitions from experi-ment and theory.”
1 J. M. Bowman, T. Carrington, and H.-D. Meyer, Mol. Phys. 106, 2145�2008�.
2 D. Lauvergnat and A. Nauts, J. Chem. Phys. 116, 8560 �2002�.3 E. Mátyus, G. Czakó, B. T. Sutcliffe, and A. G. Császár, J. Chem. Phys.
127, 084102 �2007�; E. Mátyus, J. Šimunek, and A. G. Császár, ibid.131, 074106 �2009�.
4 J. Tennyson, M. A. Kostin, P. Barletta, G. J. Harris, O. L. Polyansky, J.Ramanlal, and N. F. Zobov, Comput. Phys. Commun. 163, 85 �2004�.
5 E. Mátyus, G. Czakó, and A. G. Császár, J. Chem. Phys. 130, 134112�2009�.
6 I. N. Kozin, M. M. Law, J. Tennyson, and J. M. Hutson, Comput. Phys.Commun. 163, 117 �2004�.
7 X.-G. Wang and T. Carrington, J. Chem. Phys. 121, 2937 �2004�.8 S. N. Yurchenko, W. Thiel, and P. Jensen, J. Mol. Spectrosc. 245, 126�2007�.
9 D. Luckhaus, J. Chem. Phys. 113, 1329 �2000�.
10 C. Jung, H. S. Taylor, and M. P. Jacobson, J. Phys. Chem. A 105, 681�2001�.
11 C. Jung and H. S. Taylor, J. Phys. Chem. A 111, 3047 �2007�.12 M. E. Kellman and V. Tyng, Acc. Chem. Res. 40, 243 �2007�.13 S. C. Farantos, R. Schinke, H. Gua, and M. Joyeux, Chem. Rev. �Wash-
ington, D.C.� 109, 4248 �2009�.14 J. Tennyson, P. F. Bernath, L. R. Brown, A. Campargue, M. R. Carleer,
A. G. Császár, R. R. Gamache, J. T. Hodges, A. Jenouvrier, O. V. Nau-menko, O. L. Polyansky, L. S. Rothman, R. A. Toth, A. C. Vandaele, N.F. Zobov, L. Daumont, A. Z. Fazliev, T. Furtenbacher, I. F. Gordon, S. N.Mikhailenko, and S. V. Shirin, J. Quant. Spectrosc. Radiat. Transf. 110,573 �2009�.
15 T. Furtenbacher, A. G. Császár, and J. Tennyson, J. Mol. Spectrosc. 245,115 �2007�.
16 A. G. Császár, G. Czakó, T. Furtenbacher, and E. Mátyus, Annu. Rep.Comp. Chem. 3, 155 �2007�.
17 A. G. Császár, E. Mátyus, T. Szidarovszky, L. Lodi, N. F. Zobov, S. V.Shirin, O. L. Polyansky, and J. Tennyson, J. Quant. Spectrosc. Radiat.Transf. 111, 1043 �2010�.
18 N. E. Klepeis, A. L. L. East, A. G. Császár, W. D. Allen, T. J. Lee, and D.W. Schwenke, J. Chem. Phys. 99, 3865 �1993�.
19 T. C. Thompson and D. G. Truhlar, J. Chem. Phys. 76, 1790 �1982�.20 R. Lefebvre, Int. J. Quantum Chem. 23, 543 �1983�.21 N. de Leon and E. J. Heller, Phys. Rev. A 30, 5 �1984�.22 K. Stefański and H. S. Taylor, Phys. Rev. A 31, 2810 �1985�.23 P. R. Fleming and J. S. Hutchinson, J. Chem. Phys. 90, 1735 �1989�.24 D. W. Schwenke, J. Chem. Phys. 96, 3426 �1992�.25 R. C. Mayrhofer and E. L. Sibert, Theor. Chim. Acta 92, 107 �1995�.26 J. Zúñiga, A. Bastida, and A. Requena, J. Chem. Soc., Faraday Trans. 93,
1681 �1997�.27 J. Zúñiga, J. A. G. Picón, A. Bastida, and A. Requena, J. Chem. Phys.
122, 224319 �2005�.28 R. Dawes and T. Carrington, J. Chem. Phys. 122, 134101 �2005�.29 S. E. Choi and J. C. Light, J. Chem. Phys. 97, 7031 �1992�.30 E. B. Wilson, Jr., J. C. Decius, and P. C. Cross, Molecular Vibrations
�McGraw-Hill, New York, 1955�.31 R. N. Zare, Angular Momentum: Understanding Spatial Aspects in
Chemistry and Physics �Wiley-Interscience, New York, 1988�.
TABLE IX. RRD tables, based on Eq. �8�, for the J=3 rovibrational states of ketene �H2CCO� correspondingto the fourth pure vibrational state, showing all contributions larger than 1%.
RRD� rv , v�a,b v 603.5�B1� 603.5�B1� 437.1�B2� rv JKaKc
303�A2� 321�A2� 322�A1�
B2 607.5 100 0 0647.4 0 98 2
aSee footnote a to Table VIII.bThe variational rovibrational computations were performed as specified in footnote a to Table VII. The J=3rigid rotor eigenfunctions were constructed from the following equilibrium rotational constants: A=9.4675 cm−1, B=0.3443 cm−1, and C=0.3322 cm−1 �Ref. 79�.
034113-13 Assignment of rovibrational states J. Chem. Phys. 133, 034113 �2010�
Author complimentary copy. Redistribution subject to AIP license or copyright, see http://jcp.aip.org/jcp/copyright.jsp
32 G. D. Carney, L. I. Sprandel, and C. W. Kern, Adv. Chem. Phys. 37, 305�1978�.
33 R. J. Whitehead and N. C. Handy, Mol. Phys. 55, 456 �1975�.34 J. M. Bowman, K. M. Christoffel, and F. Tobin, J. Phys. Chem. 83, 905
�1979�.35 K. M. Dunn, J. E. Boggs, and P. Pulay, J. Chem. Phys. 85, 5838 �1986�.36 K. M. Dunn, J. E. Boggs, and P. Pulay, J. Chem. Phys. 86, 5088 �1987�.37 D. Searles and E. von Nagy-Felsobuki, J. Chem. Phys. 95, 1107 �1991�.38 J. O. Jung and R. B. Gerber, J. Chem. Phys. 105, 10332 �1996�.39 K. Yagi, T. Taketsugu, K. Hirao, and M. S. Gordon, J. Chem. Phys. 113,
1005 �2000�.40 J. M. Bowman, S. Carter, and X. Huang, Int. Rev. Phys. Chem. 22, 533
�2003�.41 T. Yonehara, T. Yamamoto, and S. Kato, Chem. Phys. Lett. 393, 98
�2004�.42 G. Rauhut, J. Chem. Phys. 121, 9313 �2004�.43 S. Carter, J. M. Bowman, and N. C. Handy, Theor. Chem. Acc. 100, 191
�1998�.44 D. Searles and E. von Nagy-Felsobuki, Ab Initio Variational Calculations
of Molecular Vibration-Rotation Spectra, Lecture Notes in Chemistry No.61 �Springer-Verlag, Berlin, 1993�.
45 O. Christiansen, Phys. Chem. Chem. Phys. 9, 2942 �2007�.46 C. Eckart, Phys. Rev. 47, 552 �1935�.47 J. K. G. Watson, Mol. Phys. 15, 479 �1968�.48 J. K. G. Watson, Mol. Phys. 19, 465 �1970�.49 D. W. Schwenke, J. Phys. Chem. 100, 2867 �1996�.50 G. Czakó, T. Furtenbacher, A. G. Császár, and V. Szalay, Mol. Phys.
102, 2411 �2004�.51 T. Furtenbacher, G. Czakó, B. T. Sutcliffe, A. G. Császár, and V. Szalay,
J. Mol. Struct. 780–781, 283 �2006�.52 M. Mladenović, Spectrochim. Acta, Part A 58, 795 �2002�.53 B. Fehrensen, D. Luckhaus, and M. Quack, Chem. Phys. Lett. 300, 312
�1999�.54 D. O. Harris, G. G. Engerholm, and W. D. Gwinn, J. Chem. Phys. 43,
1515 �1965�.55 V. Szalay, J. Chem. Phys. 99, 1978 �1993�.56 V. Szalay, G. Czakó, Á. Nagy, T. Furtenbacher, and A. G. Császár, J.
Chem. Phys. 119, 10512 �2003�.57 S. E. Choi and J. C. Light, J. Chem. Phys. 92, 2129 �1990�.58 P. R. Bunker and P. Jensen, Molecular Symmetry and Spectroscopy �NRC
Research Press, Ottawa, 1998�.59 C. Lanczos, J. Res. Natl. Bur. Stand. 45, 255 �1950�.60 X.-G. Wang and T. Carrington, Jr., J. Chem. Phys. 114, 1473 �2001�.
61 H. W. Woolley, J. Chem. Phys. 17, 347 �1949�.62 Y. Morino and K. Kuchitsu, J. Chem. Phys. 20, 1809 �1952�.63 W. J. Taylor, J. Chem. Phys. 23, 1780 �1955�.64 P. Pulay and F. Török, Acta Chir. Hung. 44, 287 �1965�.65 G. Keresztury and G. Jalsovszky, J. Mol. Struct. 10, 304 �1971�.66 W. D. Allen, A. G. Császár, and D. A. Horner, J. Am. Chem. Soc. 114,
6834 �1992�.67 G. Hose and H. S. Taylor, Phys. Rev. Lett. 51, 947 �1983�.68 V. I. Arnold, Mathematical Methods of Classical Mechanics, Graduate
Texts in Mathematics �Springer-Verlag, New York, 1978�.69 See supplementary material at http://dx.doi.org/10.1063/1.3451075 for
the definition of the normal coordinates, the reference structures used inthis study, and for numerical values used to generate Figs. 1 and 2.
70 P. Barletta, S. V. Shirin, N. F. Zobov, O. L. Polyansky, J. Tennyson, E. F.Valeev, and A. G. Császár, J. Chem. Phys. 125, 204307 �2006�.
71 O. L. Polyansky, A. G. Császár, S. V. Shirin, N. F. Zobov, P. Barletta, J.Tennyson, D. W. Schwenke, and P. J. Knowles, Science 299, 539 �2003�.
72 J. Tennyson, N. F. Zobov, R. Williamson, O. L. Polyansky, and P. F.Bernath, J. Phys. Chem. Ref. Data 30, 735 �2001�.
73 A. G. Császár and I. M. Mills, Spectrochim. Acta, Part A 53, 1101�1997�.
74 A. L. L. East, C. S. Johnson, and W. D. Allen, J. Chem. Phys. 98, 1299�1993�.
75 P. R. Schreiner, H. P. Reisenauer, F. C. Pickard, A. C. Simmonett, W. D.Allen, E. Mátyus, and A. G. Császár, Nature �London� 453, 906 �2008�.
76 P. R. Schreiner, H. P. Reisenauer, E. Mátyus, A. G. Császár, A. Siddiqi,A. C. Simmonett, and W. D. Allen, Phys. Chem. Chem. Phys. 11, 10385�2009�.
77 A. C. Simmonett, F. A. Evangelista, W. D. Allen, and H. F. Schaefer, J.Chem. Phys. 127, 014306 �2007�.
78 Y. Bing, A. C. Simmonett, G. Czakó, E. Mátyus, A. G. Császár, I. N.Kozin, and W. D. Allen �unpublished�.
79 A. L. L. East, W. D. Allen, and S. J. Klippenstein, J. Chem. Phys. 102,8506 �1995�.
80 M. Gruebele, J. W. C. Johns, and L. Nemes, J. Mol. Spectrosc. 198, 376�1999�.
81 L. Nemes, D. Luckhaus, M. Quack, and J. W. C. Johns, J. Mol. Struct.517–518, 217 �2000�.
82 J. L. Duncan, A. M. Ferguson, J. Harper, and K. H. Tonge, J. Mol.Spectrosc. 125, 196 �1987�.
83 G. Czakó, E. Mátyus, and A. G. Császár, J. Phys. Chem. A 113, 11665�2009�.
034113-14 Mátyus et al. J. Chem. Phys. 133, 034113 �2010�
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